Callaspo's three-run homer in the seventh inning gave the Angels an 8-4 lead and Shuck added two-run double later in the frame, providing what proved to be the winning margin for Los Angeles.
Mark Trumbo went 2-for-4 with a solo homer and two RBI while also scoring three times as the Angels broke a three-game losing streak.
Robert Coello (1-0) earned his first major league win with 1 2/3 scoreless innings in relief of starter Joe Blanton.
Ernesto Frieri collected his eighth save after the White Sox scored seven times late in the game.
Chicago's Nate Jones (0-4) was saddled with the loss.